thermohydrodynamic analysis of airfoil bearing based on bump foil … · 2017. 3. 2. · the...

Alexandria Engineering Journal (2016) 55, 2473–2483

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Alexandria University

Alexandria Engineering Journal

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REVIEW

Thermohydrodynamic analysis of airfoil bearingbased on bump foil structure

* Corresponding author.

E-mail address: [email protected] (S.Y. Maraiy).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

http://dx.doi.org/10.1016/j.aej.2016.06.0151110-0168 � 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

S.Y. Maraiy *, W.A. Crosby, H.A. EL-Gamal

Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt

Received 27 April 2016; revised 23 May 2016; accepted 4 June 2016

Available online 10 August 2016

KEYWORDS

Foil bearings;

Thermohydrodynamic anal-

ysis;

Bump foil structure

Abstract The load carrying capacity of the gas foil bearing depends on the material properties and

the configuration of the underlying bump strip’s structure. This paper presents three different cases

for selecting the dimensions of the foil bearing to guarantee the highest possible load carrying

capacity. It focuses on three main parameters that affect the compliance number; these parameters

are the length of bump in h direction, the pitch of bump foil, and the thickness of bump foil. It alsostudies the effect of changing these parameters on load carrying capacity according to both isother-

mal and thermohydrodynamic approaches.� 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is anopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24742. Modeling of foil support structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24753. Thermohydrodynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2476

3.1. Numerical analysis of generalized Reynolds’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24763.2. Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477

4. Different cases of selecting foil bearing dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2480

4.1. Parameters that affect the bearing performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24805. Isothermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24826. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2483

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2483

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Nomenclature

a viscosity constant of air, Pa s/�Cb bearing width, mC radial clearance, mCp specific heat at constant pressure, J/kg Kd distance between bumps, m

D bearing diameter, mDh housing diameter, me eccentricity, m

E modulus of elasticity, Pah fluid film thickness, m�h dimensionless air film thickness, hChb bump height, mhmin minimum film thickness, m�hcr dimensionless film thickness at h ¼ hcrJ number of iterations of the matlab program

K constant reflecting the structure rigidity of thebumps, m3/N

Ka conductivity of air, W/K m

l half length of bump in h direction, mL bearing length, mn number of bumps

p fluid film pressure, Pa�P dimensionless hydrodynamic pressure, �P ¼ ppapa ambient pressure, Pa

Qrec recirculating flow rate, kg/sQsuc suction flow rate, kg/sr radius of bump, mro radius of spindle, m

R radius of shaft, mS pitch of bump foil, mtb thickness of bump foil, m

T temperature of gas, �CTa ambient temperature of the air, �CTin inlet temperature of the gas film, �CTref reference temperature, �Cu linear velocity of shaft speed, m/sv linear velocity of gas flow in axial, y, direction, m/sw linear velocity of gas flow in z direction, m/s

W bearing load carrying capacity, Wx, Wz, compo-nents, N

Wt radial deformation of the bump foil due to the

steady-state aerodynamic pressure, m�Wx non dimensionless load component in the direction

of foil movement, �Wx ¼ WxPaRL�Wz non dimensionless load component in axial length

direction, �Wz ¼ WzPaRLx Cartesian coordinate in the direction of motiony Cartesian coordinate across the film thickness

z coordinate in axial length direction, m�z dimensionless coordinate in axial length direction,

�Z ¼ ZL=2

Greek symbols

a bump foil compliance numberav coefficient of cubic expansionb wrap angle, �d bearing liner deformation, me eccentricity ratio, e ¼ eCh angular coordinate in the direction of motionhcr critical value of h at subambient hydrodynamic

pressureK bearing number, K ¼ 6laxPa RC

� �2l absolute viscosity of the fluid, N s/m2

la viscosity of air, N s/m2

t Poisson ratioq density, kg/m3

u attitude angle, �w bump arc angle, �x angular velocity of the shaft, rad/s

Subscripts

a ambient, bearing entrance conditionsx; y; z quantities in the x, y, or z directionsr; h quantities in the r or h directionsv volumet time

Overbar

ðsymbolÞ non-dimensionalized parameters

2474 S.Y. Maraiy et al.

1. Introduction

In recent years, foil bearings have gained more attention than

any other types of bearings because of their unique mode ofoperation and diversity of applications. They also have variousadvantages compared to the conventional rigid journal bear-

ings in terms of higher load carrying capacity, lower powerloss, better stability, and greater endurance. These bearingsare self-acting, and can operate with ambient air or any pro-cessing gas as the lubricating fluid.

Their assembly includes a first thin smooth compliant sheetfacing the shaft, one or more corrugated foil, a second sheetbetween the foils and a compliant sheet for preventing sagging

of the first sheet between ridges of foil. Under the action of the

hydrodynamic pressure, the foil structure deforms. Therefore,

the fluid film pressure must be coupled to the deformation of the

foil structure in order to know the characteristics of the foil bear-

ing performance. From this point of view, many analytical stud-

ies have been conducted based on a range of structural models.

The concept of a foil bearing was first described in a report

over 50 years ago by Blok and Van Rossum [1]. In 1957, Patel

and Cameron [2] followed this work with another experimental

investigation using steel tape and oil by introducing a more

elaborate differential equation for finite width and derived a

less restrictive differential equation for the gap thickness than

Blok and Van Rossum [1].

Thermohydrodynamic analysis of airfoil 2475

In 1962, Gross [3] introduced the differential equations fora tape transport with air as a lubricant. In 1965, Eshel andElrod [4] rederived the differential equation of Blok and Van

Rossum [1] and presented a more refined and elaborate solu-tion for the nominal gap width than those already mentioned.They also presented numerical solutions for the film thickness

of the infinitely wide, self-acting foil bearing for various valuesof tape stiffness [5].

Eshel [6] studied the effect of compressibility on foil bear-

ings and he found that, with increasing compressibility, thenominal clearance decreased and the exit undulation decreasedin amplitude and increased in wavelength until it completelydisappeared. In 1970, he [7] also, showed that the air film,

due to self-acting lubrication effects, could be sharply reducedby small corners in the solid over which the foil passes. Heinvestigated some factors useful in overcoming excessive air

gaps in foil bearings.Heshmat et al. [8] studied the gas-lubricated foil journal

bearings, and evaluated its performance using a spring sup-

ported compliant foil as the bearing surface.In 1986, Crosby [9] made a study to understand and quan-

tify the performance of an oil-lubricated ridged foil journal

bearing of finite length and lubricated with an incompressible

Figure 1 Schematic of compliant journal bearing.

Figure 2 Single segm

fluid. He found that by decreasing the stiffness of bearing, thefilm pressure decreased and, consequently, the bearing load forthe ridged bearing was shown to be less than that for the rigid

bearing for the same eccentricity.Ku and Heshmat [10] developed a method to obtain the

stiffness of a compliant foil bearing and found that it depended

on several parameters such as the bump configuration, surfacecoating and the presence or absence of lubrication. In the samesense, Ku [11] described the effects of bearing parameters, such

as static loads, dynamics displacement amplitudes, bumps con-figurations, pivot locations and surface coating in the dynamiccharacteristics of foil bearings. In 1994, Ku and Heshmat [13]presented a theoretical model to predict the structural stiffness

and damping coefficients of the bump foil strip in a journalbearing or damper. They found, theoretically, that the energydissipated from this loop was mostly contributed by the fric-

tional motion between contact surfaces. In [14] they presentedthe results of the second part of the investigation on structuralstiffness and coulomb damping in compliant foil journal

bearings.Peng and Khonsari [17] developed a model to predict the

hydrodynamic performance of a foil journal bearing account-

ing for both the compressibility of air and the compliance ofthe bearing surface. They presented a series of predictions ofthe load-carrying capacity based on the numerical solutionfor pressure using a wide range of operating speeds. The results

showed good agreement with existing experimental data. They[18] developed a thermohydrodynamic model for predictingthe three-dimensional (3D) temperature field in an air-

lubricated, compliant foil journal bearing. The modelaccounted for the compressibility and the viscosity-temperature characteristic of air and the compliance of the

bearing surface.Kuznetsov [19] has developed a numerical THD model to

investigate the effect of lining compliance on the bearing char-

acteristics. The analysis showed increased load carrying capac-ity, significantly reduced peak pressures and thicker oil film inthe loaded zone compared to a white metal bearing. Slightlythinner oil films were predicted at the bearing edges. It was

also shown that load carrying capacity was more sensitive tothermal expansion while pressure and oil film thickness profileswere more sensitive to elastic deformation.

2. Modeling of foil support structure

The structure of bump foil bearing is shown in Fig. 1. It is

comprised of a bearing sleeve lined with corrugated bumps(bump foil), the leading edges of both bump and top foil are

ent of bump foil.


spot-welded to the bearing sleeve, the trailing edges of foils arefree, and bump foils support the single flat top foil.

The bump foils acting as springs provide stiffness, and the

smooth top foil layer provides the bearing surface. When theshaft rotates over one certain speed, the top foil expands out-ward, and the air film is generated, and the shaft is then sepa-

rated from the top foil.The elastic support structure considered in the present anal-

ysis is a simple foundation model, according to the original

work of Heshmat et al. [8] which most of the published modelsare based on. This analysis relies on several assumptions whichother researchers have also reproduced:

(1) The stiffness of a bump strip is uniformly distributedthroughout the bearing surface, i.e. the bump strip isregarded as a uniform elastic foundation.

(2) Bump stiffness is constant, independent of the actualbump deflection, not related to or constrained by adja-cent bumps.

(3) The top foil does not sag between adjacent bumps. Thetop foil does not have either bending or membrane stiff-ness, and its deflection follows that of the bump.

With these considerations, the radial deformation of thebump foil due to the steady-state aerodynamic pressure (Wt)depends on the bump compliance (a) and the average pressureacross the bearing width as shown in Fig. 2:

Wt ¼ Kðp� paÞ ð1Þwhere p and pa are the steady-state gas-film pressure and the

ambient pressure, respectively and K is a constant reflectingthe structure rigidity of the bumps. It was shown in [6] thatK is given by

K ¼ aCpa

ð2Þ

where

a ¼ 2paSCE

l

tb

� �3ð1� t2Þ ð3Þ

In order to study the performance of the foil bearing, we

should first be able to select its dimensions. Eq. (3) shows thatthere are two main parameters that affect the compliance num-ber, l the half length of bump in h direction and S the pitch ofbump foil, so we should carefully select them.

3. Thermohydrodynamic analysis

In a THD analysis, we deal with a compressible fluid; also the

energy equation and the Reynolds’ equation are coupled

Figure 3 Schematic of the region where re

through the lubricant’s viscosity-temperature relationship.These interdependences require simultaneous treatment ofboth equations to arrive at a final solution for pressure and

temperature.

3.1. Numerical analysis of generalized Reynolds’ equation

In order to study the variation of the air viscosity with temper-ature a generalized form of Reynolds’ equation is developedwhere the variation of l not only in the x and z directionsbut also in the y direction is considered.

The generalized Reynolds’ equation is as follows:

@

@xF2

ph3

l@p

@x

� �þ @@z

F2ph3

l@p

@z

� �

¼ @@x

F4 � F3Fo

� �u

� �þ @@z

F4 � F3Fo

� �w

� �ð4Þ

where

Fo ¼Z h0

dy

l; F1 ¼

Z h0

ydy

l;

F2 ¼ F1Fo

F3 �Z h0

qZ y0

ydy

ldy

F3 ¼Z h0

qZ y0

ydy

ldy;F4 ¼

Z h0

qdy

If the variation in the lubricant’s density is neglected, thegeneralized Reynolds equation in dimensionless form couldbe written in the following normalized form [20]:

@

@hI2 �P�h

3 @�P

@h

� �þ D

L

� �2@

@ �ZI2 �P�h

3 @�P

@ �Z

� �

¼ K�l @@h

�P�h 1� I1Io

� �� ð5Þ

where

Io ¼Z 10

d�y

�l; I1 ¼

Z 10

�yd�y

�l; I2 ¼

Z 10

�y

�l�y� I1

Io

� �d�y

�h ¼ 1þ e cos hþ að �P� 1ÞThe appropriate boundary conditions for the Reynolds’

equation are as follows:

�P ¼ 1 at �Z ¼ �1

@ �P

@ �Z¼ 0 at �Z ¼ 0

circulating air mixes with fresh air [18].


�Pðh ¼ 0Þ ¼ �Pðh ¼ hendÞwhere hend is the circumferential angle at which the top foilends as shown in Fig. 2.

Typically, hend ¼ 355� [60].The third boundary condition states that the pressure is

periodic in the circumferential direction.Using the finite difference method to solve Eq. (5) and sim-

plifying, the equation will be as follows:

Table 1 Bearing data.

Bearing radius, R= D/2 35 � 10�3 mBearing length, L 70 � 10�3 mBearing clearance, C 35 � 10�6 mBump foil Young’s modulus, Eb 207 � 109 N/m2Bump foil Poisson’s ratio, t 0.3

Table 2 Lubricant (air) data.

Viscosity of air, la 1:932 � 10�5 Pa sLubricant density, q 1:1614 kg/m3

Specific heat of air, Cp 1007 J/kg K

Air conductivity, Ka 2:63 � 10�2 W/m KShaft angular speed, x 30,000 rpm

�h3i;jI22

ðDhÞ2 þD

L

� �22

ðD�zÞ2" #

P2i;j þ��h3i;jI2Piþ1;j � �h3i;jI2Pi�1;j

ðDhÞ2 �D

L

� �2 �h3i;jI2Pi;jþ1 þ �h3i;jI2Pi;j�1ðD�zÞ2 þ K�l 1�

I1Io

� � �hiþ1;j � �hi�1;j2Dh

" #Pi;j

þ �h3i;jI2Piþ1;j � Pi�1;j� �2

4ðDhÞ2 �D

L

� �2�h3i;jI2

ðPi;jþ1 � Pi;j�1Þ24ðD�zÞ2 þ K�l 1�

I1Io

� ��hi;j

Piþ1;j � Pi�1;j2Dh

" #¼ 0

ð6Þ

3.2. Energy equation

The temperature distribution is determined from the energyequation. For an incompressible flow the equation is

qcp u@T

@xþ v @T

@yþ w @T

@z

� �¼ Ka @

2T

@x2þ Ka @

2T

@y2þ Ka @

2T

@z2

þ u @P@x

þ v @P@y

þ w @P@z

þ l @u@y

� �2þ @w

@y

� �2" #ð7Þ

The viscosity-temperature relationship of air is given bySalehi et al. [22]:

l ¼ aðT� TrefÞ

where a is viscosity constant of air = 4 � 10�4 Pa s/�C andTref = �458.75 �C when T is in �C.

To solve Eq. (7) we assume heat convection along h and ydirections only and assume heat conduction along y directiononly.

By using dimensionless parameters, Eq. (7) will be asfollows:

K1 �u@T

@hþ �v�h

@T

@�y

� �¼ K2 1�h2

@2T

@�y2

þ K3 �u @�P

@hþ �v 1�h

@ �P

@�yþ �w D

L

� �@ �P

@�z

� �

þ K4�l 1�h2@�u

@�y

� �2þ @ �w

@�y

� �2" #ð8Þ

where K1 ¼ qcpUTaR , K2 ¼ KaTaC2 , K3 ¼ UPaR , K4 ¼laU

2

C2.

The velocity fields in dimensionless form are given by [21]

�u ¼ 6�h2

K@�p

@h

Z �y0

�y

�ld�y� I1

Io

Z �y0

1

�ld�y

� �� 1þ 1

Io

Z �y0

1

�ld�y ð9Þ

�v ¼ ��u�l @�h

@hð10Þ

�w ¼ 6 DL

� � �h2K

@�p

@�z

Z �y0

�y

�ld�y� I1

Io

Z �y0

1

�ld�y

� �þ 1Io

Z �y0

1

�ld�y ð11Þ

The finite difference method form of Eq. (8) is

K1�u

Dhþ K1�v�hi;jDy

� K2�h2ðDyÞ2

!�Ti;k

¼ K2 1�h2�Ti;k�2 � 2 �Ti;k�1

ðDyÞ2 þK1�u

Dh�Ti�1;k þ K1�v�hi;jDy

�Ti;k�1

þ K3 �uPiþ1;j � Pi�1;j2Dh

þ �v 1�h@P

@�yþ �w D

L

� �Pi;jþ1 � Pi;j�1

2Dz

� �

þ K4�li;k 1�h2@�u

@�y

� �2þ @ �w

@�y

� �2" #

ð12ÞEq. (12) is to be solved using the following boundary

condition:

T ¼ Tin; where Tin is the inlet temperature of air �C

Figure 4 Configuration of bumps.

Figure 5 Effect of increasing number of bumps on compliance

number (Case One).

Figure 6 Effect of changing bump height on bearing compliance

number.


The parameter Tin represents the temperature at h ¼ 0,i.e., the ‘‘inlet” temperature. Since the recirculating air

temperature is generally greater than that of the fresh airentering the bearing, a mixing temperature must becalculated for Tin. The mixing temperature represents

the effective temperature after the warm air mixes withthe fresh air. A simple schematic of the control volumerepresenting the region where the air mixes is shown inFig. 3.

An energy balance in this control volume is given by

�Tin ¼�Trec �Qrec þ �Tsuc �Qsuc

�Qrec þ �Qsucð13Þ

where

�Q ¼ Q=Qref and Qref ¼PaC

3

la

D

L

� �

where �Qrec is the recirculating flow calculated using the follow-ing integration (Eq. (14)):

Figure 7 Effect of increasing bump height on load carrying capacity, e = 0.3 and rpm = 30,000.

Figure 8 Effect of increasing number of bumps on bearing load carrying capacity, rpm = 30,000, e = 0.3.


�Qrec ¼ lxL2

8PaC2

Z 10

�hcr@�y� 112

L

D

� �2 Z 10

�h3cr@ �P

@h

��h¼hcr

@�y ð14Þ

where hcr is the critical value of h where the hydrodynamicpressure becomes subambient and �hcr is the dimensionless filmthickness at h ¼ hcr.

�Qsuc is the suction flow which is equal to the side flow and iscalculated from the following integration:

�Qsuc ¼ � 112

Z hcr0

�h3@ �P

@�y

��y¼0

@h ð15Þ

Eq. (6) is solved for pressure and repeating the calculationuntil convergence is achieved for a given film thickness. Aftersome iteration using the same method of calculations and solv-

ing all the mesh points. The iterative process is carried on untilthe following convergence criterion is satisfied,

Figure 9 Effect of increasing bearing compliance number on bearing load carrying capacity, rpm = 30,000, e= 0.3.


jðP �Pi;jÞJ�1 � ðP �Pi;jÞJjjðP �Pi;jÞJj 6 10�6

After the pressure and the lubricant film profile are simulta-neously determined, numerical integration is used for the load-

carrying capacity calculation.The load carrying capacity is calculated from

�Wx ¼Z 1�1

Z hend0

ð �P� 1Þ cos h dhd�z ð16Þ

�Wz ¼Z 1�1

Z hend0

ð �P� 1Þ sin h dhd�z ð17Þ

The total load-carrying capacity is given by

�W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�W2x þ �W2z

qð18Þ

The attitude angle (u) is defined as

tanu ¼ ��Wz�Wx

Solving Eq. (12) and calculating �Ti;k, we combine it withEq. (6) using mesh size of i ¼ 60; j ¼ 20; k ¼ 20.

4. Different cases of selecting foil bearing dimensions

Tables 1 and 2 illustrate the foil bearing data used in our

calculations.The most important parameter in our study is the load car-

rying capacity, so we need to choose the dimensions that will

guarantee high values of load carrying capacity with suitableconsiderations because lightly-loaded gas bearings are veryunstable.

4.1. Parameters that affect the bearing performance

In order to design a foil bearing, we need to select the dimen-sions carefully to guarantee the highest load carrying capacity.

There are many parameters that affect the bearing perfor-mance e.g. eccentricity ratio, length to diameter ratio, compli-

ance coefficient, and bearing number.The length of the bump is considered the most important

parameter as it is the main parameter in calculating the com-pliance number a, which affects the pressure calculation thatleads to the load carrying capacity.

First, a single bump will be considered a part of a circle asshown in Fig. 4. Generally the perimeter of the full bearing is

equal to the summation of the perimeter of the whole bumpsand the spaces between them, plus the thickness of thebumps:

nð2lþ 2tb þ dÞ ¼ pðDþ 2Cþ 2hbÞ ð19ÞAlso, the pitch of the bump is calculated from Eq. (20):

S ¼ 2lþ 2tb þ d ð20Þ

In our study we will assume three different cases for the foilbearing design:

� Case One: Assuming all dimensions of the bump foil arefunctions of half bump length l.

� Case Two: Assuming variable bump height hb and the otherdimensions of the bump height are functions of half bumplength l.

� Case Three: Assuming constant housing diameter Dh andthe other dimensions of the bump height are functions ofhalf bump length l.

In Case One, we will assume the bump height, the bumpthickness, and the distance between every bump and also thepitch of bump foil as follows:

� tb ¼ 0:15 l.� hb ¼ 0:8 l.� d ¼ 0:6 l.

Figure 10 Case One: Effect of increasing number of bumps on bearing load carrying capacity e = 0.3 and rpm= 30,000.

Figure 11 Case Two: Effect of increasing number of bumps on bearing load carrying capacity, e = 0.3 and rpm= 30,000.


By substituting into Eq. (19)

l ¼ pðDþ 2CÞ2:9n� 1:6p ð21Þ

AMatlab program is used to solve Eq. (21) for values of the

number of bumps from 10 to 60 and the compliance number ais calculated.

Fig. 5 shows that the compliance number of the bearing

decreases by increasing the number of bumps.In Case Two, we will consider a variable bump height, and

the other dimensions are functions of half bump length, so the

assumptions are as follows:

� hb ¼ 1–8 mm.� d ¼ 0:6 l.� tb ¼ 0:15 l.

By substituting in Eq. (19)

l ¼ pðDþ 2Cþ 2hbÞ2:9n

ð22Þ

In order to determine the exact value of hb, we will

make the program stop when hb ¼ r and that is when2w ¼ p.

Figure 12 Case Three: Effect of increasing number of bumps on bearing load, e = 0.3 and rpm = 30,000.


as

cosw ¼ ðlþ tbÞ2 � h2b

ðlþ tbÞ2 þ h2bð23Þ

After running the program the relation between the numberof bumps, n and the compliance number, a is as shown inFig. 6.

Fig. 6 shows that by increasing the bump height, the num-

ber of bumps decreases and the bearing compliance numberalso decreases.

Eq. (6) and Eq. (12) are solved using the finite difference

method and the load carrying capacity (Eq. (18)) is calculatedusing different values of bump height. The results are as shownin Fig. 7.

Fig. 7 shows that by increasing the bump height, the dimen-sionless load carrying capacity decreases. It also reaches itshighest value with minimum bump height.

The values where hb > 5 mm are eliminated as they giveimaginary numbers of pressure. Using hb ¼ 1 mm also givesimaginary numbers of pressure, so it is better to take the rangeof hb from 2 mm to 5 mm.

Fig. 7 shows that it is better to take smaller values of bumpheight to guarantee high load carrying capacity, so hb ¼ 2 mmis the suitable value in the present design.

In Case Three, we will assume Dh ¼ 76 mm and by increas-ing the housing diameter more than the assumed value the pro-gram will stop at less number of bumps:

AsDh ¼ Dþ 2Cþ 2hb;And pDh ¼ nð2lþ 2tb þ bÞ and as in Case One

� tb ¼ 0:15 l.� hb ¼ 2 mm.� d ¼ 0:6l.

By solving the equations of the three different cases using

different values of number of bumps from 10 to 60 while

keeping other parameters constants and calculating the loadcarrying capacity using Eq. (18), the results are as shown inFig. 8.

For Case One, the load carrying capacity is increasing

with increasing the number of bumps, but in this case thereare no limitations of dimensions and the relation betweenparameters is selected by the reasonable design, also after

running the Matlab program, we started with 15 bumpsbecause using less than that number will lead to imaginaryvalues of pressure.

For Case Two, the program stopped at number of bumpsequal to 47 and by calculating the load within the ranged val-ues, and it was found that it also increases with increasing thenumber of bumps, but if we increase the number of bumps

over the ranged values, the load is still increasing, but theresults are not reasonable because at this case the bump arcangle 2W will be over 2p.

For Case Three, we calculated the load carrying capacitywithin number of bumps from 10 to 23 with constant eccentric-ity ratio = 0.3 and number of revolutions per min-

ute = 30,000. In this case the load is also increasing like thetwo previous cases, but the values are lower than the two othervalues.

Fig. 9 shows that the load carrying capacity decreases withincreasing the bearing compliance number, and increases withincreasing the number of bumps. From Eq. (3), it is shown thatthe compliance number is directly proportional to the length of

bump and inversely proportional to number of bumps asshown in Eq. (19).

5. Isothermal analysis

From the previous discussion, it is seen that using thermohy-drodynamic analysis was found to be a little complex as we

should solve Reynolds equation with the energy equation; wecan simplify the analysis by using some assumptions asfollows:


1. The change of pressure in the direction of air film thickness

is not considered.2. Gas flows only along circumferential direction.3. Viscosity of flow does not change with temperature.

According to the above assumptions, the pressure distribu-tion is expressed by Reynolds equation for compressible flow:

@

@x

ph3

l@p

@x

� �þ @@z

ph3

l@p

@z

� �¼ 6u @ðphÞ

@xð24Þ

By using dimensionless groups as follows

x ¼ Rh; @x ¼ R@h; �z ¼ zL=2

; @z ¼ L2�z; �h ¼ h

C; �P ¼ p

Pa

Eq. (24) will become

@

@h�P�h3

@ �P

@h

� �þ D

L

� �2@

@�z�P�h3

@ �P

@�z

� �¼ K @

@hð �P�hÞ ð25Þ

Using the finite difference method,Eq. (25) becomes

c1 �P2i;j þ c2 �Pi;j þ c3 ¼ 0 ð26Þ

where

c1 ¼ �h3i;j2

ðDhÞ2 þD

L

� �22

ðD�zÞ2" #

c2¼��h3i;j �Piþ1;j� �h3i;j �Pi�1;j

ðDhÞ2 �D

L

� �2 �h3i;j �Pi;jþ1þ �h3i;j �Pi;j�1ðD�zÞ2 þK

�hiþ1;j� �hi�1;j2Dh

" #

c3¼ �h3i;j�Piþ1;j� �Pi�1;j� �2

4ðDhÞ2 �D

L

� �2�h3i;j

ð �Pi;jþ1� �Pi;j�1Þ24ðD�zÞ2 þK

�hi;j�Piþ1;j� �Pi�1;j

2Dh

" #

By solving Eq. (26) for the three different cases discussedbefore, the results are shown in Figs. 10–12.

For Case One, Fig. 10 shows that the load carrying capacity

increases when considering the effect of temperature on Rey-nolds equation calculations. It also shows that the load carry-ing capacity increases more at higher number of bump.

For Case Two, Fig. 11 also shows that the load carrying

capacity is larger for the THD case compared with the isother-mal case. The load carrying capacity begins to increase more at(n > 20Þ. As we noticed before this case is reasonable in calcu-lations and for the same n, it gives higher values of load.

For Case Three, Fig. 12 shows that the load also increasesin THD case, but this case gives us less range of values of num-

ber of bumps to use in calculations.

6. Conclusion

According to the previous discussion and the comparisonbetween the isothermal and the thermohydrodynamic cases,it is clear that in order to accurately design the foil bearing,

we need to study the effect of temperature of air on the bearingperformance as an increase in gas temperature leads to anincrease in gas viscosity; hence, it affects the foil bearing per-formance. On the other hand, using isothermal approach will

help to simplify the solution and it also gives satisfactoryresults.

Also by the comparison between the three cases discussed,

it is clear that assuming a variable bump height with keepingthe other dimensions as functions of half bump length givesthe highest values of load carrying capacity and that is the

aim of the study, so Case Two is suitable for choosing thedimensions of the foil bearing. Also a wide range of numberof bumps can be used, keeping the load carrying capacity as

high as possible.

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Thermohydrodynamic analysis of airfoil bearing based on bump foil structure1 Introduction2 Modeling of foil support structure3 Thermohydrodynamic analysis3.1 Numerical analysis of generalized Reynolds’ equation3.2 Energy equation

4 Different cases of selecting foil bearing dimensions4.1 Parameters that affect the bearing performance

5 Isothermal analysis6 ConclusionReferences

thermohydrodynamic analysis of airfoil bearing based on bump foil … · 2017. 3. 2. · the...

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